Our temperature structure is based on frequency-integrated equations and supported by the grey flux-limited diffusion (FLD) simulations of
[1]}, [2]}.
FLD is an approximate method (e.g. [3]}, [4]}), but the temperature structure
should be robust because it reflects global energy conservationAs a corroboration,
in the stellar context [5]} found that non-grey collapse simulations, which also feature a shock, yielded the same structures and evolution as a frequency-averaged approach..
In particular, in our case
the temperature must drop from a value of order \(\approx T_0\)  (Equation (REF )) to a much lower value in the CSD
and therefore cross at some point this range \(T\approx 2000\) –1500 K where the opacity is particularly strong. It depends on the density only weakly (see Figure REF ),
and for a given temperature profile \(T(r)\)  and thusThis ignores the density dependence of dust opacity transitions. opacity profile \(\kappa (r)\) , \(\dot{\tau }\)  goes as \(\rho r \propto r^{-1/2}\)  (in the limit \(\zeta =1\) ; Equations (REF ) and (REF )),
which is not a strong dependence.
Thus if the temperature of maximum absorption were at another distance from the planet, the integrated absorption should be similar. The frequency dependence would be different (because of the radial dependence of the velocity) but only quantitatively.
Within the other assumptions, these results are somewhat robust since the maximum absorption does not occur in the outermost parts of the flow, where the accretion geometry is much more uncertain.
